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Mirror Equation (Spherical Mirror)

Relates the focal length of a spherical mirror to object and image distances, analogous to the thin lens equation.

Written by the Vorzep Team

OpticsGeometric OpticsMirrors

Mirror Equation CalculatorSpherical Mirror

1/f = 1/do + 1/di
f = focal length (m)  ·  do = object distance (m)  ·  di = image distance (m)
⟹ Solve⟷f, do, di
m
m
m
Please fix the errors above.
Solve for:
Presets:
Focal Length—
—
f: —do: —di: —
✓ Copied!
Mirror Type & Focal Length—
Concave (f > 0) Plane (f → ∞) Convex (f < 0)
1/f = 1/do + 1/di  ·  Sign convention: f > 0 (concave), f < 0 (convex); di > 0 (real), < 0 (virtual).
1/f = 1/d_o + 1/d_i
Mirror Equation (Spherical Mirror)

Variables

SymbolQuantityUnit
fMirror focal lengthcm
d_oObject distancecm
d_iImage distancecm

What it means

The mirror equation is identical in form to the thin lens equation, but the sign conventions differ for mirrors. For a concave mirror, f is positive; for convex, f is negative. It is used to locate images formed by spherical mirrors in telescopes, vehicle mirrors, and shaving mirrors. Understanding this equation is essential for optical design and for understanding how mirrors form images. The magnification is m = −d_i/d_o. It is also used in ray tracing for mirror systems.

Worked example

Mirror Equation – Two Detailed Examples

Real‑World
Scenario: A concave mirror with a focal length of 10 cm is used to form an image of an object placed 30 cm in front of the mirror. The student needs to find the image distance. Using 1/f = 1/d_o + 1/d_i, they compute 1/10 = 1/30 + 1/d_i → 1/d_i = 1/10 − 1/30 = 2/30 = 1/15, so d_i = 15 cm. The image is real and formed 15 cm in front of the mirror, which can be projected on a screen.
ParameterValue
do (cm)30
f (cm)10
11/d_i = 1/f − 1/d_o = 1/10 − 1/30 = (3−1)/30 = 2/30 = 1/15
2d_i = 15 cm
Result 15 cm ✓ Real image
Scenario: A convex mirror (diverging) has a focal length of −15 cm (convention: negative for convex). An object is placed 20 cm in front. The engineer wants to locate the image. Using the mirror equation, 1/f = 1/d_o + 1/d_i → 1/(−15) = 1/20 + 1/d_i → 1/d_i = −1/15 − 1/20 = −(4+3)/60 = −7/60 → d_i = −60/7 ≈ −8.57 cm. The negative sign indicates a virtual image behind the mirror, which is typical for convex mirrors, producing a smaller, upright image.
ParameterValue
do20
f−15
11/d_i = −1/15 − 1/20 = −7/60
2d_i = −60/7 ≈ −8.57 cm
Result −8.57 cm ✓ Virtual image
Insight: The mirror equation is identical in form to the thin lens equation, but sign conventions differ. For mirrors, focal length is positive for concave (converging) and negative for convex (diverging). A positive d_i indicates a real image (in front), negative indicates virtual (behind).

Common mistakes

  • Mirror equation: 1/f = 1/d_o + 1/d_i – same algebraic form as the thin lens equation.
  • Sign convention (mirror): For concave mirrors, f is positive; for convex, f is negative. d_o is positive for real objects; d_i is positive for real images (in front of mirror), negative for virtual images (behind mirror).
  • Focal length: f = R/2 for a spherical mirror – R is the radius of curvature.
  • Units: All distances in the same units.
  • Assumes: Paraxial rays (small angles) – for large apertures, spherical aberration occurs.

Applications

The mirror equation, 1/f = 1/d_o + 1/d_i, applies to spherical mirrors (concave or convex) and relates the focal length to object and image distances. This is fundamental in designing mirrors for telescopes, headlights, and reflecting optics. Engineers use it to determine image formation, to calculate the required curvature for a desired focus, and to design optical systems with mirrors. It is also used in laser cavities and in concentrating solar power systems. By applying this equation, professionals can predict the location and size of images formed by mirrors, which is essential for many practical applications. Understanding the mirror equation is crucial for anyone working with reflective optics.

  • Design of reflecting telescopes and astronomical instruments
  • Automotive headlamps and rear‑view mirrors
  • Solar concentrators and heliostats
  • Laser resonator design and optical cavities
  • Educational understanding of mirror image formation